Properties

Label 1680.2.t.h.1009.4
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.h.1009.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.23607 q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +2.23607 q^{5} +1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -4.47214i q^{13} +2.23607i q^{15} -6.47214i q^{17} +2.00000 q^{19} -1.00000 q^{21} -4.00000i q^{23} +5.00000 q^{25} -1.00000i q^{27} +8.47214 q^{29} +0.472136 q^{31} +2.00000i q^{33} +2.23607i q^{35} -2.47214i q^{37} +4.47214 q^{39} -3.52786 q^{41} +2.47214i q^{43} -2.23607 q^{45} +6.47214i q^{47} -1.00000 q^{49} +6.47214 q^{51} +2.00000i q^{53} +4.47214 q^{55} +2.00000i q^{57} -3.52786 q^{61} -1.00000i q^{63} -10.0000i q^{65} +1.52786i q^{67} +4.00000 q^{69} -12.4721 q^{71} +7.52786i q^{73} +5.00000i q^{75} +2.00000i q^{77} +8.94427 q^{79} +1.00000 q^{81} +4.94427i q^{83} -14.4721i q^{85} +8.47214i q^{87} +17.4164 q^{89} +4.47214 q^{91} +0.472136i q^{93} +4.47214 q^{95} -3.52786i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{9} + 8 q^{11} + 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} - 16 q^{31} - 32 q^{41} - 4 q^{49} + 8 q^{51} - 32 q^{61} + 16 q^{69} - 32 q^{71} + 4 q^{81} + 16 q^{89} - 8 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 4.47214i − 1.24035i −0.784465 0.620174i \(-0.787062\pi\)
0.784465 0.620174i \(-0.212938\pi\)
\(14\) 0 0
\(15\) 2.23607i 0.577350i
\(16\) 0 0
\(17\) − 6.47214i − 1.56972i −0.619671 0.784862i \(-0.712734\pi\)
0.619671 0.784862i \(-0.287266\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) 0.472136 0.0847981 0.0423991 0.999101i \(-0.486500\pi\)
0.0423991 + 0.999101i \(0.486500\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 2.23607i 0.377964i
\(36\) 0 0
\(37\) − 2.47214i − 0.406417i −0.979136 0.203208i \(-0.934863\pi\)
0.979136 0.203208i \(-0.0651369\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) −3.52786 −0.550960 −0.275480 0.961307i \(-0.588837\pi\)
−0.275480 + 0.961307i \(0.588837\pi\)
\(42\) 0 0
\(43\) 2.47214i 0.376997i 0.982073 + 0.188499i \(0.0603621\pi\)
−0.982073 + 0.188499i \(0.939638\pi\)
\(44\) 0 0
\(45\) −2.23607 −0.333333
\(46\) 0 0
\(47\) 6.47214i 0.944058i 0.881583 + 0.472029i \(0.156478\pi\)
−0.881583 + 0.472029i \(0.843522\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 6.47214 0.906280
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 4.47214 0.603023
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) − 10.0000i − 1.24035i
\(66\) 0 0
\(67\) 1.52786i 0.186658i 0.995635 + 0.0933292i \(0.0297509\pi\)
−0.995635 + 0.0933292i \(0.970249\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.4721 −1.48017 −0.740085 0.672513i \(-0.765215\pi\)
−0.740085 + 0.672513i \(0.765215\pi\)
\(72\) 0 0
\(73\) 7.52786i 0.881070i 0.897735 + 0.440535i \(0.145211\pi\)
−0.897735 + 0.440535i \(0.854789\pi\)
\(74\) 0 0
\(75\) 5.00000i 0.577350i
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 8.94427 1.00631 0.503155 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.94427i 0.542704i 0.962480 + 0.271352i \(0.0874708\pi\)
−0.962480 + 0.271352i \(0.912529\pi\)
\(84\) 0 0
\(85\) − 14.4721i − 1.56972i
\(86\) 0 0
\(87\) 8.47214i 0.908308i
\(88\) 0 0
\(89\) 17.4164 1.84614 0.923068 0.384637i \(-0.125673\pi\)
0.923068 + 0.384637i \(0.125673\pi\)
\(90\) 0 0
\(91\) 4.47214 0.468807
\(92\) 0 0
\(93\) 0.472136i 0.0489582i
\(94\) 0 0
\(95\) 4.47214 0.458831
\(96\) 0 0
\(97\) − 3.52786i − 0.358200i −0.983831 0.179100i \(-0.942681\pi\)
0.983831 0.179100i \(-0.0573186\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 4.47214 0.444994 0.222497 0.974933i \(-0.428579\pi\)
0.222497 + 0.974933i \(0.428579\pi\)
\(102\) 0 0
\(103\) − 12.9443i − 1.27544i −0.770270 0.637719i \(-0.779878\pi\)
0.770270 0.637719i \(-0.220122\pi\)
\(104\) 0 0
\(105\) −2.23607 −0.218218
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) 2.47214 0.234645
\(112\) 0 0
\(113\) 14.9443i 1.40584i 0.711269 + 0.702919i \(0.248121\pi\)
−0.711269 + 0.702919i \(0.751879\pi\)
\(114\) 0 0
\(115\) − 8.94427i − 0.834058i
\(116\) 0 0
\(117\) 4.47214i 0.413449i
\(118\) 0 0
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) − 3.52786i − 0.318097i
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) −2.47214 −0.217659
\(130\) 0 0
\(131\) −13.8885 −1.21345 −0.606724 0.794913i \(-0.707517\pi\)
−0.606724 + 0.794913i \(0.707517\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) − 2.23607i − 0.192450i
\(136\) 0 0
\(137\) 19.8885i 1.69919i 0.527433 + 0.849596i \(0.323154\pi\)
−0.527433 + 0.849596i \(0.676846\pi\)
\(138\) 0 0
\(139\) −2.94427 −0.249730 −0.124865 0.992174i \(-0.539850\pi\)
−0.124865 + 0.992174i \(0.539850\pi\)
\(140\) 0 0
\(141\) −6.47214 −0.545052
\(142\) 0 0
\(143\) − 8.94427i − 0.747958i
\(144\) 0 0
\(145\) 18.9443 1.57324
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) 12.4721 1.02176 0.510879 0.859653i \(-0.329320\pi\)
0.510879 + 0.859653i \(0.329320\pi\)
\(150\) 0 0
\(151\) 17.8885 1.45575 0.727875 0.685710i \(-0.240508\pi\)
0.727875 + 0.685710i \(0.240508\pi\)
\(152\) 0 0
\(153\) 6.47214i 0.523241i
\(154\) 0 0
\(155\) 1.05573 0.0847981
\(156\) 0 0
\(157\) − 9.41641i − 0.751511i −0.926719 0.375756i \(-0.877383\pi\)
0.926719 0.375756i \(-0.122617\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) − 23.4164i − 1.83411i −0.398755 0.917057i \(-0.630558\pi\)
0.398755 0.917057i \(-0.369442\pi\)
\(164\) 0 0
\(165\) 4.47214i 0.348155i
\(166\) 0 0
\(167\) 10.4721i 0.810358i 0.914237 + 0.405179i \(0.132791\pi\)
−0.914237 + 0.405179i \(0.867209\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 5.00000i 0.377964i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −22.9443 −1.71494 −0.857468 0.514538i \(-0.827964\pi\)
−0.857468 + 0.514538i \(0.827964\pi\)
\(180\) 0 0
\(181\) 16.4721 1.22436 0.612182 0.790717i \(-0.290292\pi\)
0.612182 + 0.790717i \(0.290292\pi\)
\(182\) 0 0
\(183\) − 3.52786i − 0.260787i
\(184\) 0 0
\(185\) − 5.52786i − 0.406417i
\(186\) 0 0
\(187\) − 12.9443i − 0.946579i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 17.4164 1.26021 0.630104 0.776511i \(-0.283012\pi\)
0.630104 + 0.776511i \(0.283012\pi\)
\(192\) 0 0
\(193\) 12.9443i 0.931749i 0.884851 + 0.465875i \(0.154260\pi\)
−0.884851 + 0.465875i \(0.845740\pi\)
\(194\) 0 0
\(195\) 10.0000 0.716115
\(196\) 0 0
\(197\) 19.8885i 1.41700i 0.705711 + 0.708500i \(0.250628\pi\)
−0.705711 + 0.708500i \(0.749372\pi\)
\(198\) 0 0
\(199\) 20.4721 1.45123 0.725616 0.688100i \(-0.241555\pi\)
0.725616 + 0.688100i \(0.241555\pi\)
\(200\) 0 0
\(201\) −1.52786 −0.107767
\(202\) 0 0
\(203\) 8.47214i 0.594627i
\(204\) 0 0
\(205\) −7.88854 −0.550960
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −25.8885 −1.78224 −0.891120 0.453767i \(-0.850080\pi\)
−0.891120 + 0.453767i \(0.850080\pi\)
\(212\) 0 0
\(213\) − 12.4721i − 0.854577i
\(214\) 0 0
\(215\) 5.52786i 0.376997i
\(216\) 0 0
\(217\) 0.472136i 0.0320507i
\(218\) 0 0
\(219\) −7.52786 −0.508686
\(220\) 0 0
\(221\) −28.9443 −1.94700
\(222\) 0 0
\(223\) − 20.9443i − 1.40253i −0.712900 0.701266i \(-0.752618\pi\)
0.712900 0.701266i \(-0.247382\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) − 9.88854i − 0.656326i −0.944621 0.328163i \(-0.893571\pi\)
0.944621 0.328163i \(-0.106429\pi\)
\(228\) 0 0
\(229\) 26.3607 1.74196 0.870981 0.491316i \(-0.163484\pi\)
0.870981 + 0.491316i \(0.163484\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) − 2.94427i − 0.192886i −0.995339 0.0964428i \(-0.969254\pi\)
0.995339 0.0964428i \(-0.0307465\pi\)
\(234\) 0 0
\(235\) 14.4721i 0.944058i
\(236\) 0 0
\(237\) 8.94427i 0.580993i
\(238\) 0 0
\(239\) 15.5279 1.00441 0.502207 0.864747i \(-0.332522\pi\)
0.502207 + 0.864747i \(0.332522\pi\)
\(240\) 0 0
\(241\) −11.8885 −0.765808 −0.382904 0.923788i \(-0.625076\pi\)
−0.382904 + 0.923788i \(0.625076\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −2.23607 −0.142857
\(246\) 0 0
\(247\) − 8.94427i − 0.569110i
\(248\) 0 0
\(249\) −4.94427 −0.313331
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) 0 0
\(255\) 14.4721 0.906280
\(256\) 0 0
\(257\) − 23.4164i − 1.46068i −0.683086 0.730338i \(-0.739363\pi\)
0.683086 0.730338i \(-0.260637\pi\)
\(258\) 0 0
\(259\) 2.47214 0.153611
\(260\) 0 0
\(261\) −8.47214 −0.524412
\(262\) 0 0
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 0 0
\(265\) 4.47214i 0.274721i
\(266\) 0 0
\(267\) 17.4164i 1.06587i
\(268\) 0 0
\(269\) −30.3607 −1.85112 −0.925562 0.378597i \(-0.876407\pi\)
−0.925562 + 0.378597i \(0.876407\pi\)
\(270\) 0 0
\(271\) −24.4721 −1.48658 −0.743288 0.668971i \(-0.766735\pi\)
−0.743288 + 0.668971i \(0.766735\pi\)
\(272\) 0 0
\(273\) 4.47214i 0.270666i
\(274\) 0 0
\(275\) 10.0000 0.603023
\(276\) 0 0
\(277\) − 23.4164i − 1.40696i −0.710717 0.703478i \(-0.751629\pi\)
0.710717 0.703478i \(-0.248371\pi\)
\(278\) 0 0
\(279\) −0.472136 −0.0282660
\(280\) 0 0
\(281\) 2.94427 0.175641 0.0878203 0.996136i \(-0.472010\pi\)
0.0878203 + 0.996136i \(0.472010\pi\)
\(282\) 0 0
\(283\) − 16.9443i − 1.00723i −0.863927 0.503616i \(-0.832003\pi\)
0.863927 0.503616i \(-0.167997\pi\)
\(284\) 0 0
\(285\) 4.47214i 0.264906i
\(286\) 0 0
\(287\) − 3.52786i − 0.208243i
\(288\) 0 0
\(289\) −24.8885 −1.46403
\(290\) 0 0
\(291\) 3.52786 0.206807
\(292\) 0 0
\(293\) 24.9443i 1.45726i 0.684908 + 0.728630i \(0.259843\pi\)
−0.684908 + 0.728630i \(0.740157\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.00000i − 0.116052i
\(298\) 0 0
\(299\) −17.8885 −1.03452
\(300\) 0 0
\(301\) −2.47214 −0.142492
\(302\) 0 0
\(303\) 4.47214i 0.256917i
\(304\) 0 0
\(305\) −7.88854 −0.451697
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 12.9443 0.736374
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) − 25.4164i − 1.43662i −0.695723 0.718310i \(-0.744916\pi\)
0.695723 0.718310i \(-0.255084\pi\)
\(314\) 0 0
\(315\) − 2.23607i − 0.125988i
\(316\) 0 0
\(317\) − 6.94427i − 0.390029i −0.980800 0.195015i \(-0.937525\pi\)
0.980800 0.195015i \(-0.0624754\pi\)
\(318\) 0 0
\(319\) 16.9443 0.948697
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 12.9443i − 0.720239i
\(324\) 0 0
\(325\) − 22.3607i − 1.24035i
\(326\) 0 0
\(327\) − 14.9443i − 0.826420i
\(328\) 0 0
\(329\) −6.47214 −0.356820
\(330\) 0 0
\(331\) −33.8885 −1.86268 −0.931341 0.364147i \(-0.881361\pi\)
−0.931341 + 0.364147i \(0.881361\pi\)
\(332\) 0 0
\(333\) 2.47214i 0.135472i
\(334\) 0 0
\(335\) 3.41641i 0.186658i
\(336\) 0 0
\(337\) 11.0557i 0.602244i 0.953586 + 0.301122i \(0.0973611\pi\)
−0.953586 + 0.301122i \(0.902639\pi\)
\(338\) 0 0
\(339\) −14.9443 −0.811661
\(340\) 0 0
\(341\) 0.944272 0.0511352
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 8.94427 0.481543
\(346\) 0 0
\(347\) 30.8328i 1.65519i 0.561324 + 0.827596i \(0.310292\pi\)
−0.561324 + 0.827596i \(0.689708\pi\)
\(348\) 0 0
\(349\) −25.4164 −1.36051 −0.680255 0.732976i \(-0.738131\pi\)
−0.680255 + 0.732976i \(0.738131\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) 17.5279i 0.932914i 0.884544 + 0.466457i \(0.154470\pi\)
−0.884544 + 0.466457i \(0.845530\pi\)
\(354\) 0 0
\(355\) −27.8885 −1.48017
\(356\) 0 0
\(357\) 6.47214i 0.342542i
\(358\) 0 0
\(359\) 13.4164 0.708091 0.354045 0.935228i \(-0.384806\pi\)
0.354045 + 0.935228i \(0.384806\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 16.8328i 0.881070i
\(366\) 0 0
\(367\) − 25.8885i − 1.35137i −0.737190 0.675685i \(-0.763848\pi\)
0.737190 0.675685i \(-0.236152\pi\)
\(368\) 0 0
\(369\) 3.52786 0.183653
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) − 16.3607i − 0.847124i −0.905867 0.423562i \(-0.860780\pi\)
0.905867 0.423562i \(-0.139220\pi\)
\(374\) 0 0
\(375\) 11.1803i 0.577350i
\(376\) 0 0
\(377\) − 37.8885i − 1.95136i
\(378\) 0 0
\(379\) −5.88854 −0.302474 −0.151237 0.988498i \(-0.548326\pi\)
−0.151237 + 0.988498i \(0.548326\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 20.3607i 1.04038i 0.854050 + 0.520191i \(0.174139\pi\)
−0.854050 + 0.520191i \(0.825861\pi\)
\(384\) 0 0
\(385\) 4.47214i 0.227921i
\(386\) 0 0
\(387\) − 2.47214i − 0.125666i
\(388\) 0 0
\(389\) 2.58359 0.130993 0.0654967 0.997853i \(-0.479137\pi\)
0.0654967 + 0.997853i \(0.479137\pi\)
\(390\) 0 0
\(391\) −25.8885 −1.30924
\(392\) 0 0
\(393\) − 13.8885i − 0.700584i
\(394\) 0 0
\(395\) 20.0000 1.00631
\(396\) 0 0
\(397\) 7.52786i 0.377813i 0.981995 + 0.188906i \(0.0604943\pi\)
−0.981995 + 0.188906i \(0.939506\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −15.8885 −0.793436 −0.396718 0.917941i \(-0.629851\pi\)
−0.396718 + 0.917941i \(0.629851\pi\)
\(402\) 0 0
\(403\) − 2.11146i − 0.105179i
\(404\) 0 0
\(405\) 2.23607 0.111111
\(406\) 0 0
\(407\) − 4.94427i − 0.245078i
\(408\) 0 0
\(409\) 7.88854 0.390063 0.195032 0.980797i \(-0.437519\pi\)
0.195032 + 0.980797i \(0.437519\pi\)
\(410\) 0 0
\(411\) −19.8885 −0.981030
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 11.0557i 0.542704i
\(416\) 0 0
\(417\) − 2.94427i − 0.144182i
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −10.9443 −0.533391 −0.266696 0.963781i \(-0.585932\pi\)
−0.266696 + 0.963781i \(0.585932\pi\)
\(422\) 0 0
\(423\) − 6.47214i − 0.314686i
\(424\) 0 0
\(425\) − 32.3607i − 1.56972i
\(426\) 0 0
\(427\) − 3.52786i − 0.170725i
\(428\) 0 0
\(429\) 8.94427 0.431834
\(430\) 0 0
\(431\) 21.4164 1.03159 0.515796 0.856711i \(-0.327496\pi\)
0.515796 + 0.856711i \(0.327496\pi\)
\(432\) 0 0
\(433\) 21.4164i 1.02921i 0.857428 + 0.514603i \(0.172061\pi\)
−0.857428 + 0.514603i \(0.827939\pi\)
\(434\) 0 0
\(435\) 18.9443i 0.908308i
\(436\) 0 0
\(437\) − 8.00000i − 0.382692i
\(438\) 0 0
\(439\) 31.5279 1.50474 0.752371 0.658739i \(-0.228910\pi\)
0.752371 + 0.658739i \(0.228910\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 36.9443i 1.75528i 0.479325 + 0.877638i \(0.340882\pi\)
−0.479325 + 0.877638i \(0.659118\pi\)
\(444\) 0 0
\(445\) 38.9443 1.84614
\(446\) 0 0
\(447\) 12.4721i 0.589912i
\(448\) 0 0
\(449\) 9.05573 0.427366 0.213683 0.976903i \(-0.431454\pi\)
0.213683 + 0.976903i \(0.431454\pi\)
\(450\) 0 0
\(451\) −7.05573 −0.332241
\(452\) 0 0
\(453\) 17.8885i 0.840477i
\(454\) 0 0
\(455\) 10.0000 0.468807
\(456\) 0 0
\(457\) 8.94427i 0.418395i 0.977873 + 0.209198i \(0.0670852\pi\)
−0.977873 + 0.209198i \(0.932915\pi\)
\(458\) 0 0
\(459\) −6.47214 −0.302093
\(460\) 0 0
\(461\) −12.4721 −0.580885 −0.290443 0.956892i \(-0.593803\pi\)
−0.290443 + 0.956892i \(0.593803\pi\)
\(462\) 0 0
\(463\) 34.8328i 1.61882i 0.587245 + 0.809409i \(0.300212\pi\)
−0.587245 + 0.809409i \(0.699788\pi\)
\(464\) 0 0
\(465\) 1.05573i 0.0489582i
\(466\) 0 0
\(467\) − 19.0557i − 0.881794i −0.897558 0.440897i \(-0.854660\pi\)
0.897558 0.440897i \(-0.145340\pi\)
\(468\) 0 0
\(469\) −1.52786 −0.0705502
\(470\) 0 0
\(471\) 9.41641 0.433885
\(472\) 0 0
\(473\) 4.94427i 0.227338i
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −11.0557 −0.504098
\(482\) 0 0
\(483\) 4.00000i 0.182006i
\(484\) 0 0
\(485\) − 7.88854i − 0.358200i
\(486\) 0 0
\(487\) 13.8885i 0.629350i 0.949199 + 0.314675i \(0.101896\pi\)
−0.949199 + 0.314675i \(0.898104\pi\)
\(488\) 0 0
\(489\) 23.4164 1.05893
\(490\) 0 0
\(491\) −1.05573 −0.0476443 −0.0238222 0.999716i \(-0.507584\pi\)
−0.0238222 + 0.999716i \(0.507584\pi\)
\(492\) 0 0
\(493\) − 54.8328i − 2.46955i
\(494\) 0 0
\(495\) −4.47214 −0.201008
\(496\) 0 0
\(497\) − 12.4721i − 0.559452i
\(498\) 0 0
\(499\) −5.88854 −0.263607 −0.131804 0.991276i \(-0.542077\pi\)
−0.131804 + 0.991276i \(0.542077\pi\)
\(500\) 0 0
\(501\) −10.4721 −0.467861
\(502\) 0 0
\(503\) 39.4164i 1.75749i 0.477291 + 0.878745i \(0.341619\pi\)
−0.477291 + 0.878745i \(0.658381\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) − 7.00000i − 0.310881i
\(508\) 0 0
\(509\) 10.5836 0.469109 0.234555 0.972103i \(-0.424637\pi\)
0.234555 + 0.972103i \(0.424637\pi\)
\(510\) 0 0
\(511\) −7.52786 −0.333013
\(512\) 0 0
\(513\) − 2.00000i − 0.0883022i
\(514\) 0 0
\(515\) − 28.9443i − 1.27544i
\(516\) 0 0
\(517\) 12.9443i 0.569288i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −9.41641 −0.412540 −0.206270 0.978495i \(-0.566133\pi\)
−0.206270 + 0.978495i \(0.566133\pi\)
\(522\) 0 0
\(523\) 7.05573i 0.308525i 0.988030 + 0.154263i \(0.0493002\pi\)
−0.988030 + 0.154263i \(0.950700\pi\)
\(524\) 0 0
\(525\) −5.00000 −0.218218
\(526\) 0 0
\(527\) − 3.05573i − 0.133110i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.7771i 0.683382i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 22.9443i − 0.990118i
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −22.9443 −0.986451 −0.493226 0.869901i \(-0.664182\pi\)
−0.493226 + 0.869901i \(0.664182\pi\)
\(542\) 0 0
\(543\) 16.4721i 0.706887i
\(544\) 0 0
\(545\) −33.4164 −1.43140
\(546\) 0 0
\(547\) 37.3050i 1.59504i 0.603289 + 0.797522i \(0.293856\pi\)
−0.603289 + 0.797522i \(0.706144\pi\)
\(548\) 0 0
\(549\) 3.52786 0.150566
\(550\) 0 0
\(551\) 16.9443 0.721850
\(552\) 0 0
\(553\) 8.94427i 0.380349i
\(554\) 0 0
\(555\) 5.52786 0.234645
\(556\) 0 0
\(557\) − 5.05573i − 0.214218i −0.994247 0.107109i \(-0.965841\pi\)
0.994247 0.107109i \(-0.0341594\pi\)
\(558\) 0 0
\(559\) 11.0557 0.467607
\(560\) 0 0
\(561\) 12.9443 0.546508
\(562\) 0 0
\(563\) − 34.8328i − 1.46803i −0.679134 0.734014i \(-0.737645\pi\)
0.679134 0.734014i \(-0.262355\pi\)
\(564\) 0 0
\(565\) 33.4164i 1.40584i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 10.1115 0.423151 0.211576 0.977362i \(-0.432141\pi\)
0.211576 + 0.977362i \(0.432141\pi\)
\(572\) 0 0
\(573\) 17.4164i 0.727581i
\(574\) 0 0
\(575\) − 20.0000i − 0.834058i
\(576\) 0 0
\(577\) − 24.4721i − 1.01879i −0.860533 0.509394i \(-0.829870\pi\)
0.860533 0.509394i \(-0.170130\pi\)
\(578\) 0 0
\(579\) −12.9443 −0.537946
\(580\) 0 0
\(581\) −4.94427 −0.205123
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 10.0000i 0.413449i
\(586\) 0 0
\(587\) − 15.0557i − 0.621416i −0.950505 0.310708i \(-0.899434\pi\)
0.950505 0.310708i \(-0.100566\pi\)
\(588\) 0 0
\(589\) 0.944272 0.0389080
\(590\) 0 0
\(591\) −19.8885 −0.818105
\(592\) 0 0
\(593\) − 21.3050i − 0.874890i −0.899245 0.437445i \(-0.855884\pi\)
0.899245 0.437445i \(-0.144116\pi\)
\(594\) 0 0
\(595\) 14.4721 0.593300
\(596\) 0 0
\(597\) 20.4721i 0.837869i
\(598\) 0 0
\(599\) 18.3607 0.750197 0.375099 0.926985i \(-0.377609\pi\)
0.375099 + 0.926985i \(0.377609\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) − 1.52786i − 0.0622194i
\(604\) 0 0
\(605\) −15.6525 −0.636364
\(606\) 0 0
\(607\) − 4.94427i − 0.200682i −0.994953 0.100341i \(-0.968007\pi\)
0.994953 0.100341i \(-0.0319933\pi\)
\(608\) 0 0
\(609\) −8.47214 −0.343308
\(610\) 0 0
\(611\) 28.9443 1.17096
\(612\) 0 0
\(613\) − 7.41641i − 0.299546i −0.988720 0.149773i \(-0.952146\pi\)
0.988720 0.149773i \(-0.0478543\pi\)
\(614\) 0 0
\(615\) − 7.88854i − 0.318097i
\(616\) 0 0
\(617\) 47.8885i 1.92792i 0.266047 + 0.963960i \(0.414282\pi\)
−0.266047 + 0.963960i \(0.585718\pi\)
\(618\) 0 0
\(619\) 11.8885 0.477841 0.238920 0.971039i \(-0.423206\pi\)
0.238920 + 0.971039i \(0.423206\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 17.4164i 0.697774i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 4.00000i 0.159745i
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −23.0557 −0.917834 −0.458917 0.888479i \(-0.651762\pi\)
−0.458917 + 0.888479i \(0.651762\pi\)
\(632\) 0 0
\(633\) − 25.8885i − 1.02898i
\(634\) 0 0
\(635\) 8.94427i 0.354943i
\(636\) 0 0
\(637\) 4.47214i 0.177192i
\(638\) 0 0
\(639\) 12.4721 0.493390
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 7.05573i 0.278251i 0.990275 + 0.139125i \(0.0444291\pi\)
−0.990275 + 0.139125i \(0.955571\pi\)
\(644\) 0 0
\(645\) −5.52786 −0.217659
\(646\) 0 0
\(647\) − 38.2492i − 1.50373i −0.659316 0.751866i \(-0.729154\pi\)
0.659316 0.751866i \(-0.270846\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.472136 −0.0185045
\(652\) 0 0
\(653\) 1.05573i 0.0413138i 0.999787 + 0.0206569i \(0.00657577\pi\)
−0.999787 + 0.0206569i \(0.993424\pi\)
\(654\) 0 0
\(655\) −31.0557 −1.21345
\(656\) 0 0
\(657\) − 7.52786i − 0.293690i
\(658\) 0 0
\(659\) −12.1115 −0.471795 −0.235898 0.971778i \(-0.575803\pi\)
−0.235898 + 0.971778i \(0.575803\pi\)
\(660\) 0 0
\(661\) 48.2492 1.87668 0.938339 0.345717i \(-0.112364\pi\)
0.938339 + 0.345717i \(0.112364\pi\)
\(662\) 0 0
\(663\) − 28.9443i − 1.12410i
\(664\) 0 0
\(665\) 4.47214i 0.173422i
\(666\) 0 0
\(667\) − 33.8885i − 1.31217i
\(668\) 0 0
\(669\) 20.9443 0.809752
\(670\) 0 0
\(671\) −7.05573 −0.272383
\(672\) 0 0
\(673\) 21.8885i 0.843741i 0.906656 + 0.421871i \(0.138626\pi\)
−0.906656 + 0.421871i \(0.861374\pi\)
\(674\) 0 0
\(675\) − 5.00000i − 0.192450i
\(676\) 0 0
\(677\) − 2.11146i − 0.0811499i −0.999176 0.0405749i \(-0.987081\pi\)
0.999176 0.0405749i \(-0.0129189\pi\)
\(678\) 0 0
\(679\) 3.52786 0.135387
\(680\) 0 0
\(681\) 9.88854 0.378930
\(682\) 0 0
\(683\) 14.1115i 0.539960i 0.962866 + 0.269980i \(0.0870171\pi\)
−0.962866 + 0.269980i \(0.912983\pi\)
\(684\) 0 0
\(685\) 44.4721i 1.69919i
\(686\) 0 0
\(687\) 26.3607i 1.00572i
\(688\) 0 0
\(689\) 8.94427 0.340750
\(690\) 0 0
\(691\) 48.8328 1.85769 0.928844 0.370471i \(-0.120804\pi\)
0.928844 + 0.370471i \(0.120804\pi\)
\(692\) 0 0
\(693\) − 2.00000i − 0.0759737i
\(694\) 0 0
\(695\) −6.58359 −0.249730
\(696\) 0 0
\(697\) 22.8328i 0.864855i
\(698\) 0 0
\(699\) 2.94427 0.111363
\(700\) 0 0
\(701\) −0.472136 −0.0178323 −0.00891616 0.999960i \(-0.502838\pi\)
−0.00891616 + 0.999960i \(0.502838\pi\)
\(702\) 0 0
\(703\) − 4.94427i − 0.186477i
\(704\) 0 0
\(705\) −14.4721 −0.545052
\(706\) 0 0
\(707\) 4.47214i 0.168192i
\(708\) 0 0
\(709\) −35.8885 −1.34782 −0.673911 0.738812i \(-0.735387\pi\)
−0.673911 + 0.738812i \(0.735387\pi\)
\(710\) 0 0
\(711\) −8.94427 −0.335436
\(712\) 0 0
\(713\) − 1.88854i − 0.0707265i
\(714\) 0 0
\(715\) − 20.0000i − 0.747958i
\(716\) 0 0
\(717\) 15.5279i 0.579899i
\(718\) 0 0
\(719\) 8.94427 0.333565 0.166783 0.985994i \(-0.446662\pi\)
0.166783 + 0.985994i \(0.446662\pi\)
\(720\) 0 0
\(721\) 12.9443 0.482070
\(722\) 0 0
\(723\) − 11.8885i − 0.442140i
\(724\) 0 0
\(725\) 42.3607 1.57324
\(726\) 0 0
\(727\) − 19.0557i − 0.706738i −0.935484 0.353369i \(-0.885036\pi\)
0.935484 0.353369i \(-0.114964\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) − 5.63932i − 0.208293i −0.994562 0.104147i \(-0.966789\pi\)
0.994562 0.104147i \(-0.0332111\pi\)
\(734\) 0 0
\(735\) − 2.23607i − 0.0824786i
\(736\) 0 0
\(737\) 3.05573i 0.112559i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 8.94427 0.328576
\(742\) 0 0
\(743\) − 2.83282i − 0.103926i −0.998649 0.0519630i \(-0.983452\pi\)
0.998649 0.0519630i \(-0.0165478\pi\)
\(744\) 0 0
\(745\) 27.8885 1.02176
\(746\) 0 0
\(747\) − 4.94427i − 0.180901i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.8328 −0.979143 −0.489572 0.871963i \(-0.662847\pi\)
−0.489572 + 0.871963i \(0.662847\pi\)
\(752\) 0 0
\(753\) − 24.0000i − 0.874609i
\(754\) 0 0
\(755\) 40.0000 1.45575
\(756\) 0 0
\(757\) 30.4721i 1.10753i 0.832673 + 0.553764i \(0.186809\pi\)
−0.832673 + 0.553764i \(0.813191\pi\)
\(758\) 0 0
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −24.4721 −0.887114 −0.443557 0.896246i \(-0.646284\pi\)
−0.443557 + 0.896246i \(0.646284\pi\)
\(762\) 0 0
\(763\) − 14.9443i − 0.541019i
\(764\) 0 0
\(765\) 14.4721i 0.523241i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.05573 −0.0380705 −0.0190353 0.999819i \(-0.506059\pi\)
−0.0190353 + 0.999819i \(0.506059\pi\)
\(770\) 0 0
\(771\) 23.4164 0.843321
\(772\) 0 0
\(773\) − 16.9443i − 0.609443i −0.952441 0.304722i \(-0.901437\pi\)
0.952441 0.304722i \(-0.0985634\pi\)
\(774\) 0 0
\(775\) 2.36068 0.0847981
\(776\) 0 0
\(777\) 2.47214i 0.0886874i
\(778\) 0 0
\(779\) −7.05573 −0.252798
\(780\) 0 0
\(781\) −24.9443 −0.892576
\(782\) 0 0
\(783\) − 8.47214i − 0.302769i
\(784\) 0 0
\(785\) − 21.0557i − 0.751511i
\(786\) 0 0
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) 0 0
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −14.9443 −0.531357
\(792\) 0 0
\(793\) 15.7771i 0.560261i
\(794\) 0 0
\(795\) −4.47214 −0.158610
\(796\) 0 0
\(797\) − 28.0000i − 0.991811i −0.868377 0.495905i \(-0.834836\pi\)
0.868377 0.495905i \(-0.165164\pi\)
\(798\) 0 0
\(799\) 41.8885 1.48191
\(800\) 0 0
\(801\) −17.4164 −0.615379
\(802\) 0 0
\(803\) 15.0557i 0.531305i
\(804\) 0 0
\(805\) 8.94427 0.315244
\(806\) 0 0
\(807\) − 30.3607i − 1.06875i
\(808\) 0 0
\(809\) 24.8328 0.873075 0.436538 0.899686i \(-0.356205\pi\)
0.436538 + 0.899686i \(0.356205\pi\)
\(810\) 0 0
\(811\) −13.0557 −0.458449 −0.229224 0.973374i \(-0.573619\pi\)
−0.229224 + 0.973374i \(0.573619\pi\)
\(812\) 0 0
\(813\) − 24.4721i − 0.858275i
\(814\) 0 0
\(815\) − 52.3607i − 1.83411i
\(816\) 0 0
\(817\) 4.94427i 0.172978i
\(818\) 0 0
\(819\) −4.47214 −0.156269
\(820\) 0 0
\(821\) 19.3050 0.673747 0.336874 0.941550i \(-0.390630\pi\)
0.336874 + 0.941550i \(0.390630\pi\)
\(822\) 0 0
\(823\) − 15.0557i − 0.524810i −0.964958 0.262405i \(-0.915484\pi\)
0.964958 0.262405i \(-0.0845156\pi\)
\(824\) 0 0
\(825\) 10.0000i 0.348155i
\(826\) 0 0
\(827\) − 32.7214i − 1.13783i −0.822395 0.568917i \(-0.807363\pi\)
0.822395 0.568917i \(-0.192637\pi\)
\(828\) 0 0
\(829\) −3.52786 −0.122528 −0.0612639 0.998122i \(-0.519513\pi\)
−0.0612639 + 0.998122i \(0.519513\pi\)
\(830\) 0 0
\(831\) 23.4164 0.812306
\(832\) 0 0
\(833\) 6.47214i 0.224246i
\(834\) 0 0
\(835\) 23.4164i 0.810358i
\(836\) 0 0
\(837\) − 0.472136i − 0.0163194i
\(838\) 0 0
\(839\) 32.9443 1.13736 0.568681 0.822558i \(-0.307454\pi\)
0.568681 + 0.822558i \(0.307454\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 0 0
\(843\) 2.94427i 0.101406i
\(844\) 0 0
\(845\) −15.6525 −0.538462
\(846\) 0 0
\(847\) − 7.00000i − 0.240523i
\(848\) 0 0
\(849\) 16.9443 0.581526
\(850\) 0 0
\(851\) −9.88854 −0.338975
\(852\) 0 0
\(853\) − 31.5279i − 1.07949i −0.841827 0.539747i \(-0.818520\pi\)
0.841827 0.539747i \(-0.181480\pi\)
\(854\) 0 0
\(855\) −4.47214 −0.152944
\(856\) 0 0
\(857\) 16.5836i 0.566485i 0.959048 + 0.283242i \(0.0914101\pi\)
−0.959048 + 0.283242i \(0.908590\pi\)
\(858\) 0 0
\(859\) −21.0557 −0.718412 −0.359206 0.933258i \(-0.616952\pi\)
−0.359206 + 0.933258i \(0.616952\pi\)
\(860\) 0 0
\(861\) 3.52786 0.120229
\(862\) 0 0
\(863\) − 18.8328i − 0.641077i −0.947236 0.320538i \(-0.896136\pi\)
0.947236 0.320538i \(-0.103864\pi\)
\(864\) 0 0
\(865\) 26.8328i 0.912343i
\(866\) 0 0
\(867\) − 24.8885i − 0.845259i
\(868\) 0 0
\(869\) 17.8885 0.606827
\(870\) 0 0
\(871\) 6.83282 0.231521
\(872\) 0 0
\(873\) 3.52786i 0.119400i
\(874\) 0 0
\(875\) 11.1803i 0.377964i
\(876\) 0 0
\(877\) − 49.3050i − 1.66491i −0.554093 0.832455i \(-0.686935\pi\)
0.554093 0.832455i \(-0.313065\pi\)
\(878\) 0 0
\(879\) −24.9443 −0.841349
\(880\) 0 0
\(881\) 27.5279 0.927437 0.463719 0.885983i \(-0.346515\pi\)
0.463719 + 0.885983i \(0.346515\pi\)
\(882\) 0 0
\(883\) 16.3607i 0.550581i 0.961361 + 0.275290i \(0.0887740\pi\)
−0.961361 + 0.275290i \(0.911226\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 4.58359i − 0.153902i −0.997035 0.0769510i \(-0.975482\pi\)
0.997035 0.0769510i \(-0.0245185\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 12.9443i 0.433164i
\(894\) 0 0
\(895\) −51.3050 −1.71494
\(896\) 0 0
\(897\) − 17.8885i − 0.597281i
\(898\) 0 0
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 12.9443 0.431236
\(902\) 0 0
\(903\) − 2.47214i − 0.0822675i
\(904\) 0 0
\(905\) 36.8328 1.22436
\(906\) 0 0
\(907\) − 16.3607i − 0.543247i −0.962404 0.271624i \(-0.912439\pi\)
0.962404 0.271624i \(-0.0875606\pi\)
\(908\) 0 0
\(909\) −4.47214 −0.148331
\(910\) 0 0
\(911\) 25.4164 0.842083 0.421042 0.907041i \(-0.361665\pi\)
0.421042 + 0.907041i \(0.361665\pi\)
\(912\) 0 0
\(913\) 9.88854i 0.327263i
\(914\) 0 0
\(915\) − 7.88854i − 0.260787i
\(916\) 0 0
\(917\) − 13.8885i − 0.458640i
\(918\) 0 0
\(919\) −36.7214 −1.21133 −0.605663 0.795721i \(-0.707092\pi\)
−0.605663 + 0.795721i \(0.707092\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 55.7771i 1.83593i
\(924\) 0 0
\(925\) − 12.3607i − 0.406417i
\(926\) 0 0
\(927\) 12.9443i 0.425146i
\(928\) 0 0
\(929\) −19.3050 −0.633375 −0.316687 0.948530i \(-0.602571\pi\)
−0.316687 + 0.948530i \(0.602571\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) − 16.0000i − 0.523816i
\(934\) 0 0
\(935\) − 28.9443i − 0.946579i
\(936\) 0 0
\(937\) 15.3050i 0.499991i 0.968247 + 0.249995i \(0.0804291\pi\)
−0.968247 + 0.249995i \(0.919571\pi\)
\(938\) 0 0
\(939\) 25.4164 0.829433
\(940\) 0 0
\(941\) 35.5279 1.15818 0.579088 0.815265i \(-0.303409\pi\)
0.579088 + 0.815265i \(0.303409\pi\)
\(942\) 0 0
\(943\) 14.1115i 0.459532i
\(944\) 0 0
\(945\) 2.23607 0.0727393
\(946\) 0 0
\(947\) − 16.0000i − 0.519930i −0.965618 0.259965i \(-0.916289\pi\)
0.965618 0.259965i \(-0.0837111\pi\)
\(948\) 0 0
\(949\) 33.6656 1.09283
\(950\) 0 0
\(951\) 6.94427 0.225183
\(952\) 0 0
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 0 0
\(955\) 38.9443 1.26021
\(956\) 0 0
\(957\) 16.9443i 0.547731i
\(958\) 0 0
\(959\) −19.8885 −0.642235
\(960\) 0 0
\(961\) −30.7771 −0.992809
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 28.9443i 0.931749i
\(966\) 0 0
\(967\) 25.8885i 0.832519i 0.909246 + 0.416260i \(0.136659\pi\)
−0.909246 + 0.416260i \(0.863341\pi\)
\(968\) 0 0
\(969\) 12.9443 0.415830
\(970\) 0 0
\(971\) 33.8885 1.08754 0.543768 0.839236i \(-0.316997\pi\)
0.543768 + 0.839236i \(0.316997\pi\)
\(972\) 0 0
\(973\) − 2.94427i − 0.0943890i
\(974\) 0 0
\(975\) 22.3607 0.716115
\(976\) 0 0
\(977\) 52.8328i 1.69027i 0.534552 + 0.845136i \(0.320480\pi\)
−0.534552 + 0.845136i \(0.679520\pi\)
\(978\) 0 0
\(979\) 34.8328 1.11326
\(980\) 0 0
\(981\) 14.9443 0.477134
\(982\) 0 0
\(983\) 25.5279i 0.814212i 0.913381 + 0.407106i \(0.133462\pi\)
−0.913381 + 0.407106i \(0.866538\pi\)
\(984\) 0 0
\(985\) 44.4721i 1.41700i
\(986\) 0 0
\(987\) − 6.47214i − 0.206010i
\(988\) 0 0
\(989\) 9.88854 0.314437
\(990\) 0 0
\(991\) 56.9443 1.80889 0.904447 0.426586i \(-0.140284\pi\)
0.904447 + 0.426586i \(0.140284\pi\)
\(992\) 0 0
\(993\) − 33.8885i − 1.07542i
\(994\) 0 0
\(995\) 45.7771 1.45123
\(996\) 0 0
\(997\) 5.63932i 0.178599i 0.996005 + 0.0892995i \(0.0284628\pi\)
−0.996005 + 0.0892995i \(0.971537\pi\)
\(998\) 0 0
\(999\) −2.47214 −0.0782149
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1680.2.t.h.1009.4 4
3.2 odd 2 5040.2.t.u.1009.2 4
4.3 odd 2 840.2.t.c.169.2 4
5.2 odd 4 8400.2.a.db.1.1 2
5.3 odd 4 8400.2.a.cz.1.2 2
5.4 even 2 inner 1680.2.t.h.1009.2 4
12.11 even 2 2520.2.t.f.1009.1 4
15.14 odd 2 5040.2.t.u.1009.1 4
20.3 even 4 4200.2.a.bk.1.2 2
20.7 even 4 4200.2.a.bj.1.1 2
20.19 odd 2 840.2.t.c.169.4 yes 4
60.59 even 2 2520.2.t.f.1009.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.c.169.2 4 4.3 odd 2
840.2.t.c.169.4 yes 4 20.19 odd 2
1680.2.t.h.1009.2 4 5.4 even 2 inner
1680.2.t.h.1009.4 4 1.1 even 1 trivial
2520.2.t.f.1009.1 4 12.11 even 2
2520.2.t.f.1009.2 4 60.59 even 2
4200.2.a.bj.1.1 2 20.7 even 4
4200.2.a.bk.1.2 2 20.3 even 4
5040.2.t.u.1009.1 4 15.14 odd 2
5040.2.t.u.1009.2 4 3.2 odd 2
8400.2.a.cz.1.2 2 5.3 odd 4
8400.2.a.db.1.1 2 5.2 odd 4